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Casimir interaction between a plate and a cylinder T. Emig,a R. L. Jaffe,b,c M. Kardar,c and A. Scardicchiob,c aInstitut fu¨r Theoretische Physik, Universität zu Köln, Zu¨lpicher Straße 77, 50937 Köln, Germany 6 bCenter for Theoretical Physics and Laboratory for Nuclear Science 0 cDepartment of Physics 0 2 Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Dated: February 6, 2008) n a WefindtheexactCasimirforcebetweenaplateandacylinder,ageometryintermediatebetween J parallel plates, where the force is known exactly, and the plate–sphere, where it is known at large 3 separations. The force has an unexpectedly weak decay ∼L/(H3ln(H/R)) at large plate–cylinder separations H (L and R are the cylinder length and radius), due to transverse magnetic modes. ] Path integral quantization with a partial wave expansion additionally gives a qualitative difference h for thedensity of states of electric and magnetic modes, and corrections at finitetemperatures. c e m PACSnumbers: 42.25.Fx,03.70.+k,12.20.-m - t a With recent advances in the fabrication of electronic a plate and a sphere of radius R, E ~cR/H2 [9] for st and mechanical systems on the nanometer scale quan- the Dirichlet case, as compared with E∼ ~cR3/H4 for . tum effects like Casimir forces have become increasingly the EM case [6]. Based on these result∼s, expectations t a important[1,2]. Thesesystemscanprobemechanicalos- for the plate and cylinder geometry might range from m cillation modes of quasi one-dimensional structures such ~cLR2/H4, proportional to the cylinder volume, to - asnanowiresorcarbonnanotubeswithhighprecision[3]. ∼ ~cLR/H3, proportional to its surface area, or even d n However, thorough theoretical investigations of Casimir ∼ ~cL/H2 with a potential non-power law dependence ∼ o forces are to date limited to “closed” geometries such as on the radius. c parallel plates [4] or, recently, a rectilinear “piston” [5], A simple but uncontrolled method for study of non- [ where the zero point fluctuations are not diffracted into planar geometries is the proximity force approximation 1 regionswhichareinaccessibletoclassicalrays. Anotable (PFA), where the systemis treatedas a sumof infinites- v exception is the original work by Casimir and Polder on mal parallel plates [10]. Applied to the plate–cylinder 5 the interaction between a plate and an atom (sphere) at geometry, the PFA yields E = 1 π3~cL R/2a5 5 PFA −960 asymptotically large separation [6]. to leading order in a/R, where a = H R.pOther 0 − 1 In this Letter we consider the electrodynamic Casimir approximations include semi-classical methods based on 0 the Gutzwiller trace formula [11], and a recent optical interaction between a plate and a parallel cylinder (or 6 approach which sums also over closed but non-periodic “wire”), both assumed to be perfect metals (see inset of 0 paths[12]. For large separations, a multiple scattering / Fig. 1). We show that the Casimir interaction can be t approach is available [13], but has not been adapted to a computed without approximation for this geometry. We m believe that the methods presentedhere may yield exact this geometry. For the Dirichlet case, a Monte Carlo approach based on worldline techniques has been applied - solutions for other interesting geometries as well. This d geometryisalsoofrecentexperimentalinterest: Keeping to the plate-cylinder case [14]. n Our result provides a test for the validity of these ap- two plates parallel has proved very difficult. The sphere o proximate schemes, and also provides insight into the andplateconfigurationavoidsthisproblem,buttheforce c large distance limit. In particular, we find the unex- : is not extensive. The cylinder is easier to hold parallel v and the force is extensive in its length [7]. pected result that the electrodynamic Casimir force for i X the plate–cylinder geometry has the weakest of the pos- Casimir interactions, while attractive for perfect met- sible decays, r als in all known cases, depend strongly on geometry. a Considerthe Casimir interactionenergy(discarding sep- 1 ~cL F = , (1) aration independent terms) at asymptotically large H −8πH3ln(H/R) for three fundamental geometries which differ in the co- dimension of the surfaces[8]: two plates, plate–cylinder, as H/R . The same asymptotic result applies to a → ∞ and finally, plate–sphere, corresponding to co-dimension scalarfieldwith Dirichletboundaryconditions. Interest- 1, 2, and 3, respectively. It is instructive to consider ingly, the decay exponent of the force is not monotonic both a scalar field which vanishes on the surfaces (D inthe numberofco-dimensions: ( 4, 3 ǫ, 5)forco- ≡ − − − − Dirichlet) and the electromagnetic field (EM). For par- dimension (1,2,3) respectively. In contrast the Dirichlet allel plates (area A) E ~cA/H3 in both cases [4]. For case is monotonic with exponents ( 4, 3 ǫ, 3). ∼ − − − − 2 In the remainder we derive these results, summarized can be expressed as, in Eqs. (5)-(8), using path integral techniques. Our ap- ~cL detM˜(q ,q ) proach also yields the distance dependent part of the 0 1 E = dq dq ln . (3) density of states, which contains the complete geometry 8π2 ZZ 0 1 detM˜ (q ,q ) ∞ 0 1 dependent information of the photon spectrum, and is The elements of the matrix M are labeled by the in- useful for computing thermal contributions to the force. teger index m = ,..., coming from the compact Thetranslationalsymmetryalongthecylinderaxisen- ϕ-dimensionofthe−c∞ylinder∞,andthemomentumq along 2 ables a decomposition of the EM field into transverse the other direction parallel to the plate, to read magnetic (TM) and electric (TE) modes [15] which are described by a scalar field obeying Dirichlet (D) or Neu- M˜ = A[m,m′] B[m,q2′] . (4) mann (N) boundary conditions respectively. We can (cid:18)B[Tq2,m′] C[q2,q2′] (cid:19) compare our TM results to recent Monte Carlo Dirich- letresults[14]. Moreover,themodedecompositionturns The matrix A[m,m′] is diagonal, with elements A A = I (ru)K (ru) for D and out to be useful also in identifying the physical mecha- [m,m] m m m ≡ A = (u/H)2I′ (ru)K′ (ru) for N modes. The ma- nism behind the weak decay of the force, which at large m m m trix C also has only diagonal elements C distance is fully dominated by D modes. [q2,q2] ≡ C(q ) = H/(2 u2+u2) for D and C(q ) = Ourstartingpointisapathintegralrepresentation[16] 2 2 2 u2+u2/(2H) pfor N modes. The off-diagonal ma- for the effective action which yields a trace formula for − 2 trpix B is non-diagonal with B B (q ) = the density of states (DOS) [17]. The latter is then eval- [m,q2] ≡ m 2 uated using a partial wave expansion. The DOS on the πHe−√u2+u22Im(ru)[u/(u2+ u2+u22)]m/ u2+u22 for ibmyaρg(iinqa0r)y=fre(q2uq0e/nπcy) axdi3sxisGr(exla,txe;dq0t)o,awGherreeenG’s(xfu,nxc′;tiqo0n) Dua2n+du2B)m]−(mq2)for=N m(πodue/sH.p)He−er√e,u2w+eu22hIam′v(perud)e[fiun/e(du2th+e 2 is the Green’s functRion for the scalar field with action pdimensionless combinations u = H q2+q2, u = Hq S = 1 d3x( Φ2+(q0)2Φ2). The effect of boundaries 0 1 2 2 2 |∇ | and r = R/H. The determinapnt can be obtained on theRGreen function can be obtained by placing func- straightforwardly, and the total energy can be decom- tional delta functions on the boundary surfaces in the posed to the sum of D and N mode contributions, as functional integral[16]. By integrating out both the field Φandtheauxiliaryfieldswhichrepresentthedeltafunc- ~cL E = ΦD(r)+ΦN(r) , (5) tions on the surfaces, one obtains the trace formula [17] −H2 (cid:2) (cid:3) with 1 ∞ 1 ∂ ΦX(r)= duuln det( NX(u,r)) . (6) δρ(q0)=−π∂q0Trln(cid:0)MM∞−1(cid:1) , (2) −4π Z0 (cid:2) 1− (cid:3) ThematrixNX(u)isgivenintermsofBesselfunctions, where δρ(q ) is the change in the DOS caused by mov- 0 I (ru) ing the plate and cylinder in from infinity. The infor- ND(u,r)= ν K (2u), (7) µν K (ru) µ+ν mation about geometry is contained in the matrix M µ of the quadratic action for the auxiliary fields, given by for D modes and M (u,u′;q ) = G (s (u) s (u′);q ) for D and by NMααbββo(uun,dua′;ryq00)co=nd∂intiαo0(nus)α;∂nGβ(u−′=)Geβ0−(sqα0|(xu|/)04−πsxβ(uis′)t;hqe0)frfoeer NµNν(u,r)=−KIν′µ′((rruu))Kµ+ν(2u), (8) 0 | | space Green function, ∂nα is its derivative normal to the for N modes. The determinant in Eq. (6) is taken with surface, and s (u) parametrizes the surfaces (which are α respect to the integer indices µ, ν = ,..., . If the numbered by α = 1, 2) in terms of surface coordinates matrix NX is restricted to dimension−(∞2l+1)∞with NX u. M−1 is the functional inverse of M at infinite sur- 00 ∞ asthe centralelement, itthendescribes the contribution face separation. The trace in Eq. (2) runs over u and from l partial waves, beginning with s-waves for l=0. α. For the cylinder with its axis oriented along the x 1 From Eq. (6), one can easily extract the asymptotic direction we set s (x ,ϕ) = (x ,Rsin(ϕ),Rcos(ϕ)) and 1 1 1 large distance behavior of the energy for r = R/H 1. forthe plates (x ,x )=(x ,x ,H)(seeinsetofFig.1). ≪ 2 1 2 1 2 For Dirichlet modes s-waves dominate, while for Neu- The Casimir energy of interaction is given by E = mann modes both s- and p-waves (l = 1) contribute at (~c/2) ∞dq q δρ(q ). After transforming to momen- leadingorderinr. Thetwocasesdifferqualitatively,with 0 0 0 0 tum spRace, M˜, the Fourier transform of the matrix M has block diagonal form with respect to q0 and the mo- 1 1 5 mentumq alongthecylinderaxis,sotheCasimirenergy ΦD(r)= , and ΦN(r)= r2. (9) 1 −16πlnr 32π 3 For H R the EM Casimir interaction is dominated ≫ by the D (TM) modes. Note that a naive application of 2R the PFA for small r, where it is not justified, yields the TM (Dirichlet) incorrect scalings ΦD(r)=ΦN(r) r. ∼ a The natural expectation from the Casimir–Polder result for the plate-sphere interaction, that the force at large distance is proportionalto the volume of the cylin- H A der, is incorrect. The physical reason for this difference F P E is explained by considering spontaneous charge fluctua- E/ tions. On a sphere, the positive and negative charges can be separated by at most distances of order R EM ≪ H. The retarded van der Waals interactions between these dipoles and their images on the plate leads to the Casimir–Polder interaction [6]. In the cylinder, fluctuations of charge along the axis of the cylinder can create TE (Neumann) arbitrarylargepositively(ornegatively)chargedregions. The retarded interaction of these charges (not dipoles) a/R withtheirimagesgivesthedominanttermoftheCasimir FIG. 1: Ratio of the exact Casimir energy to the PFA for force. Thisinterpretationisconsistentwiththedifference the geometry shown in the inset. All curves are obtained at between the two types of modes, since for N modes such order 25 of thepartial wave expansion, and theaccuracy lies charge modulations cannot occur due to the absence of within the line thickness even at small a/R. The Dirichlet anelectricfieldalongthe cylinder axis. Eventually,for a data points are from Ref. [19]. finite cylinder, in the very far region H L, the charge ≫ fluctuationscanbeconsideredagainassmalldipoles,and the Casimir–Polder law is expected to reappear, making Eq. (2) we obtain the expression theforceproportionaltothevolumeofthecylinderLR2. Wenextconsiderarbitraryseparations,anduseEq.(6) q HL ∞du det NX( u2+(q0H)2) to obtain the contribution from higher order partial δρX(q0)=− 0π2 Z u2 ln h1d−et[ pNX(q H)] i, 0 1− 0 waves. A numerical evaluation of the determinant is (10) straightforward, and we find that down to even small whichisconvenientbothfornumericalandanalyticcom- separations of a/R = 0.1 the energy converges at order putations. Numericalevaluationyieldstheresultsshown l = 25, whereas for a/R & 1 convergence is achieved for in Fig. 2 for general values of R. Analytical results in l=4. Fig.1showsourresultsforDirichletandNeumann the limit of small R/H are obtained by considering only modesandfortheirsumwhichistheEMCasimirenergy, the s-waves for Dirichlet modes, and the s- and p-waves all scaled by the corresponding E given above [18]. for Neumann modes. For Dirichlet modes we expand in PFA Both types of modes show a strong deviation from the 1/ln(q R), whereas for Neumann modes the small pa- 0 PFA for a/R&1, especially the Dirichlet energy. Fig. 1 rameter is r =R/H. To leading order we find shows also very recent wordline-based Monte Carlo results for the Dirichlet case at moderate separations [19], L e−2q0H δρ (q )= + ln−2(q R) , (11a) which agree nicely with our exact results. D 0 2πln(q R) O 0 0 (cid:0) (cid:1) Eq. (9) indicates that the Dirichlet dominated force δρ (q )= q0HL(1+6q H)e−2q0Hr2+ (r3).(11b) N 0 0 vanishes logarithmically as R 0 at fixed H. A similar − 8π O → resultis obtainedwhenthe cylinderis replacedbyanin- Fig.2allowsforanassessmentofthevalidityrangeofthe finitesimal thin wire, but an UV cutoff is introduced to expansions of Eq. (11) which are shown as solid curves. control short wavelength modes. Both results are a con- These results for the DOS allow us to evaluate for the sequence of the fact that the asymptotic form of Eq. (9) firsttimefinitetemperaturecontributionstotheCasimir is independent of the actual shape of the cross section interactionin anopen geometry. The difference between of the wire, and the cutoff R can be identified with any the free energy and the Casimir energy at T = 0 can typical scale of the cross section. The leading asymp- F be written as [13] (k is Boltzmann constant) totic term in Eq.(9) is also obtained[8] fromthe s-wave B scattering amplitude for the 2-dimensional problem of a ∞ strongly repulsive potential concentrated on the wire. δ = E =πkBT dq0g(q0)δρ(q0), (12) F F − Z 0 The difference between the D and N modes also ap- pears in the density of states, which in turn affects the with the function g(q ) = ∞ sin(2πkq λ )/(πk) and 0 k=1 0 T temperature dependence of the Casimir force. From λT = ~c/(2πkBT). In thePlimit R (H,λT) but for ≪ 4 general H/λ , we can use the expansion of Eq. (11) to T (a) a/R=0.1 (l=25) obtain to leading order in 1/ln(R/λ ) and R/H for D T and N modes, respectively, the thermal contributions k T L H λ B T δ = coth , (13a) D F 8 ln(R/λ )H (cid:20) (cid:18)λ (cid:19)− H (cid:21) T T k T Lλ R2 H H B T δ = 7 coth (13b) FN − 64 H4 (cid:20) λT (cid:18)λT(cid:19) a/R=1.0 (l=4) 7(H/λ )2 H 3 cosh(H/λ ) T T + +6 20 . sinh2(H/λT) (cid:18)λT(cid:19) sinh3(H/λT) − (cid:21) It is interesting to note that δ has a minimum at N a/R=10.0 (l=2) F H/λ =2.915...,wherethecorrespondingthermalforce T changes from repulsive at small H to attractive at large q a 0 H. At low temperatures, the finite T contributions to the Casimir force δF = ∂δ /∂H, (b) − F a/R=0.1 (l=25) 2π3 k T 3 HL B δFD = 45 kBT (cid:18) ~c (cid:19) ln(R/λ ), (14a) a/R=1.0 (l=4) T 64π5 k T 5 δF = k T B R2HL, (14b) N 945 B (cid:18) ~c (cid:19) have to be added to Eq. (1) for H λ . At larger T ≪ temperatures with R λ H, one has the scalings T ≪ ≪ δFD kBTLH−2/ln(R/λT)andδFN kBTLR2H−4. a/R=10.0 (l=2) ∼ ∼− At the extreme high temperature limit of λ R, only T thermalfluctuationsremain,and~shoulddisap≪pearfrom the equations. This ‘classical limit’ is well known for parallel plates [20] and is obtained for smooth, arbitrary q a 0 geometries within the multiple scattering approach [13], FIG. 2: The change in thedensity of states for (a) Dirichlet andthe opticalapproximation[21]. (Note thatfor the D and (b) Neumann modes obtained from Eq. (10) at order l. modes a subleading ~ still survives in the logarithm.) The solid curves show thesmall R expansion of Eq. (11). Finally, we note that our approach can be extended [8] For a recent study of the co-dimension dependence see also to multiple wires and distorted beams. Our re- A. Scardicchio, Phys.Rev.D 72, 065004 (2005). sults should be relevant to nano-systems composed of [9] A. Bulgac, P. Magierski, and A. Wirzba, Preprint 1-dimensionalstructures andalsoto other types offields hep-th/0511056. as, e.g., thermal order parameter fluctuations. [10] M. Bordag, U. Mohideen, V. M. Mostepanenko, Phys. WethankH.Giesfordiscussionsandespeciallyforpro- Rep. 353, 1 (2001). viding the data of Fig. 1 prior to publication. This work [11] M. Schaden and L. Spruch, Phys. Rev. Lett. 84, 459 (2000). wassupportedbytheDFGthroughgrantEM70/2(TE), [12] R. L. Jaffe and A. Scardicchio, Phys. Rev. Lett. 92 the Istituto Nazionale di Fisica Nulceare (AS), the NSF 070402 (2004); A. Scardicchio and R. L. Jaffe, Nucl. through grant DMR-04-26677 (MK), and the U. S. De- Phys. B 704, 552 (2005) [arXiv:quant-ph/0406041]. partmentofEnergy(D.O.E.)undercooperativeresearch [13] R. Balian and B. Duplantier, Ann. Phys. (New York) agreement #DF-FC02-94ER40818(RLJ & AS). 104, 300 (1977); 112, 165 (1978). [14] H. Gies, K. Langfeld, and L. Moyaerts, JHEP 0306 018 [1] A. N. Cleland and M. L. Roukes, Appl. Phys. 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